Prime geodesic theorem for the modular surface

Year 2020, , 505 - 509, 02.04.2020

Muharem Avdispahić

https://doi.org/10.15672/hujms.568323

Cited By: 2

Abstract

Under the generalized Lindelöf hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon$ outside a set of finite logarithmic measure.

Keywords

prime geodesic theorem, Selberg zeta function, modular group

References

• [1] M. Avdispahić, Prime geodesic theorem of Gallagher type, arXiv:1701.02115, 2017.
• [2] M. Avdispahić, On Koyama’s refinement of the prime geodesic theorem, Proc. Japan Acad. Ser. A Math. Sci. 94 (3), 21–24, 2018.
• [3] M. Avdispahić, Gallagherian $PGT$ on $PSL(2,Z)$, Funct. Approx. Comment. Math. 58 (2), 207–213, 2018.
• [4] M. Avdispahić, Errata and addendum to "On the prime geodesic theorem for hyperbolic 3-manifolds" Math. Nachr. 291 (2018), no. 14-15, 2160–2167, Math. Nachr. 292 (4), 691–693, 2019.
• [5] O. Balkanova and D. Frolenkov, Bounds for a spectral exponential sum, J. London Math. Soc. 99 (2), 249–272, 2019.
• [6] R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: With applications to string theory, in: Lect. Notes Phys. 779, Springer, Berlin Heidelberg, 2009.
• [7] P. Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathe- matics 106, Birkhäuser, Boston-Basel-Berlin, 1992.
• [8] Y. Cai, Prime geodesic theorem, J. Théor. Nombres Bordeaux, 14 (1), 59–72, 2002.
• [9] J.B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic L-functions, Ann. of Math. (2) 151 (3), 1175–1216, 2000.
• [10] P.X. Gallagher, A large sieve density estimate near $\sigma =1$, Invent. Math. 11, 329–339, 1970.
• [11] P.X. Gallagher, Some consequences of the Riemann hypothesis, Acta Arith. 37, 339– 343, 1980.
• [12] D.A. Hejhal, The Selberg trace formula for PSL(2,R). Vol I, Lecture Notes in Math- ematics 548, Springer, Berlin, 1976.
• [13] A.E. Ingham, The distribution of prime numbers, Cambridge University Press, 1932.
• [14] H. Iwaniec, Non-holomorphic modular forms and their applications, in: Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., 157– 196, Horwood, Chichester, 1984.
• [15] H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349, 136–159, 1984.
• [16] S. Koyama, Refinement of prime geodesic theorem, Proc. Japan Acad. Ser A Math. Sci. 92 (7), 77–81, 2016.
• [17] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on $PSL_{2}(Z)\backslash H^{2}$, Inst. Hautes Études Sci. Publ. Math. 81, 207–237, 1995.
• [18] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233, 241–247, 1977.
• [19] K. Soundararajan and M.P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676, 105–120, 2013.